Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,11,16,9,15)(7,18,22,12,23,32,21,31)(8,24,26,14,28,13,27,33)(19,30,36,29,20,34,25,35);(1,2)(3,4)(5,7,11,23)(6,12,16,31)(8,25,28,36)(9,22,10,21)(13,30,24,34)(14,29,33,35)(15,32,17,18)(19,26,20,27);(2,4)(5,8)(6,13)(7,19)(9,26)(10,27)(11,28)(12,29)(14,17)(15,33)(16,24)(18,34)(20,23)(21,36)(22,25)(30,32)(31,35);(5,8)(6,14)(7,20)(9,27)(10,26)(11,28)(12,30)(13,17)(15,24)(16,33)(18,35)(19,23)(21,36)(22,25)(29,32)(31,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265644, #72265649, #72265675, … |
| maximal normal subgroup of | #72265644, #72265649, #72265675, … |
| maximal quotient of | #72238924 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290256);